Generalized Liénard systems, singularly perturbed systems, Flow Curvature Method

Abstract

In his famous book entitled \textit{Theory of Oscillations}, Nicolas Minorsky wrote: "\textit{each time the system absorbs energy the curvature of its trajectory decreases} and \textit{vice versa}". According to the \textit{Flow Curvature Method}, the location of the points where the \textit{curvature of trajectory curve}, integral of such planar \textit{singularly dynamical systems}, vanishes directly provides a first order approximation in $\epsilon$ of its \textit{slow invariant manifold} equation. By using this method, we prove that, in the $\epsilon$-vicinity of the \textit{slow invariant manifold} of generalized Liénard systems, the \textit{curvature of trajectory curve} increases while the \textit{energy} of such systems decreases. Hence, we prove Minorsky's statement for the generalized Liénard systems. Then, we establish a relationship between \textit{curvature} and \textit{energy} for such systems. These results are then exemplified with the classical Van der Pol and generalized Liénard \textit{singularly perturbed systems}.

19 pages, 1 figure. arXiv admin note: text overlap with arXiv:1408.4894

References in corpus (2)

• Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems
• Canards from Chua's circuit